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On the -Theory of Finite Fields ON THE 퐾-THEORY OF FINITE FIELDS PETER J. HAINE Abstract. In this talk we introduce higher algebraic 퐾-groups via Quillen’s plus construc- tion. We then give a brief tour of algebraic 퐾-theory and its relation to stable homotopy the- ory and number theory — in particular noting that a 퐾-theory computation of the integers is equivalent to the Kummer–Vandiver conjecture. We next state Quillen’s computations of the 퐾-theory of 퐅푞 and 퐅푞. The remainder of the talk focuses on how Quillen is able to execute these 퐾-theory computations by using the Adams operations to relate 퐾-theory to modular representation theory. We finish by discussing the Brauer lift to — the main tool that makes Quillen’s computation go — and how Quillen uses the Brauer lift to show that + 푞 퐵GL(퐅푞) is homotopy equivalent to a space 퐹휓 , defined via the Adams operations, whose homotopy groups are easily computable using Bott periodicity. Contents 1. Algebraic 퐾-theory via perfect groups and the Plus Construction 1 2. A Whirlwind Tour of 퐾-theory 4 3. The 퐾-theory of finite fields 4 4. The Adams Operations via 휆-rings 5 5. The Brauer Lift 8 6. The space 퐹휓푞 and computing the 퐾-theory of finite fields 9 References 11 1. Algebraic 퐾-theory via perfect groups and the Plus Construction In this section we define higher algebraic 퐾-theory via Quillen’s plus construction. To motivate this, we recall how the group 퐾1 was first defined via elementary matrices. 1.1. Convention. For us 푅 always denotes a commutative (unital) ring. 1.2. Recollection. Let 푛 be a positive integer, and consider the group GL푛(푅) of 푛 × 푛 ma- trices. Given 1 ≤ 푖, 푗 ≤ 푛 and 푟 ∈ 푅, let 훿푖,푗 by the 푛 × 푛 matrix whose only nonzero entry is 1 in the (푖, 푗) position, and let 휀푖,푗(푟) ≔ id푛 +푟훿푖,푗. The matrices 휀푖,푗(푟) are called elementary matrices. Let 퐸푛(푅) denote the subgroup of GL푛(푅) generated by the elementary matrices. The infinite general linear group is the colimit GL(푅) ≔ colim ( GL1(푅) GL2(푅) GL2(푅) ⋯ ) . Date: Fall 2016. Acknowledgments: We thank Robin Elliott for watching a preliminary version of this talk and for providing extremely helpful suggestions on its contents and organization. Note: Items marked with a “ * ” are extra remarks that we likely did not have time to mention in the talk. 1 2 PETER J. HAINE Similarly, define 퐸(푅) ≔ colim ( 퐸1(푅) 퐸2(푅) 퐸2(푅) ⋯ ) . 1.3. Remark. If 푅 is a field, then 퐸푛(푅) = SL푛(푅). 1.1. Preliminary Definition. Let 푅 be a commutative ring. The first algebraic 퐾-group of 푅 is the quotient 퐾1(푅) ≔ GL(푅)/퐸(푅) . As algebraic topologists, we prefer definitions to have a homotopy-theoretic flavor. In particular, we might try to define 퐾1(푅) in terms of homotopy groups. We already know that 휋1(퐵GL(푅)) ≅ GL(푅). Moreover, 퐵GL(푅) has an explicit CW-structure. One idea to produce 퐾1 homotopy–theoretically is to add cells to 퐵GL(푅) to create a space whose funda- mental group is 퐾1(푅) — is is precisely what the plus construction will do for us. Before we do this, let us take a closer look at 퐸(푅) to see exactly what sort of group-theoretic properties 퐸(푅) has. 1.4. Notation. Let 퐺 be a group. Write [퐺, 퐺] for the commutator subgroup of 퐺 and write 퐺ab ≅ 퐺/[퐺, 퐺] for the abelianization of 퐺. 1.5. Example. For 푛 ≥ 3, the subgroup 퐸푛(푅) is a normal subgroup of GL푛(푅) with the property that [퐸푛(푅), 퐸푛(푅)] = 퐸푛(푅). Moreover, [퐸(푅), 퐸(푅)] = 퐸(푅). 1.6. Definition. A group 퐺 is perfect if 퐺ab = 0. 1.7. Example. The alternating group 퐴5 ⊲ 훴5 is the smallest nontrivial perfect group. 1.8. Example. The infinite symmetric group is the colimit 훴∞ ≔ colim ( 훴1 훴2 훴2 ⋯ ) . The infinite alternating group is the colimit 퐴∞ ≔ colim ( 퐴1 퐴2 퐴2 ⋯ ) . The index of 퐴∞ in 훴∞ is 2, and 퐴∞ is a perfect normal subgroup of 훴∞. Acyclic spaces generate a class of examples of perfect groups. ̃ 1.9. Definition. A space 푋 is acyclic if 퐻∗(푋; 퐙) = 0. 1.10. Lemma. Let 푋 be an acyclic space. Then 푋 is connected and 휋1(푋) is perfect. Proof. Since 푋 is acyclic, 퐻0(푋) ≅ 퐙, hence 푋 is connected. Similarly, since 푋 is acyclic 퐻1(푋) = 0, but the Hurewicz homomorphism exhibits 퐻1(푋) as an abelianization of 휋1(푋), □ hence 휋1(푋)ab = 0. 1.11. Definition. Let 푋 and 푌 be pointed connected CW-complexes. A pointed map 푓∶ 푋 → 푌 is acyclic if hofib(푓) is acyclic. 1.12. Lemma. Let 푋 and 푌 be connected CW-complexes. A map 푓∶ 푋 → 푌 is acyclic if and only if for every local coefficient system 퐿∶ 훱1(푌) → Ab, the map 푓 induces an isomorphism 푓⋆ ∶ 퐻∗(푋; 퐿 ∘ 훱1(푓)) ⥲ 퐻∗(푌; 퐿) . ON THE 퐾-THEORY OF FINITE FIELDS 3 Proof Sketch. The proof is not very hard — the main tool of is the Serre spectral sequence, and the comparison theorem for cellular maps of fibrations. A proof can be found in [13, Ch. 4 Lem. 1.6]. □ 1.13. Definition. Let 푋 be a pointed connected CW-complex and 푃 ⊲ 휋1(푋) a perfect normal subgroup. An acyclic map 휄∶ 푋 → 푋+ is called a plus construction for 푋 (relative + to 푃) if 푃 is the kernel of 휄⋆ ∶ 휋1(푋) → 휋1(푋 ). 1.14. Theorem (Quillen). Let 푋 be a pointed CW-complex and 푃 ⊲ 휋1(푋) a perfect normal subgroup. Then there exists a plus construction 휄∶ 푋 → 푋+ for 푋 relative to 푃. Moreover, the plus construction enjoys the following universal property: for every map 푓∶ 푋 → 푌 so that + + 푓⋆(푃) = 0, there is a map 푓 ∶ 푋 → 푌, unique up to pointed homotopy, so that the triangle 푋 휄 푋+ ≃ 푓 푓+ 푌 commutes up to pointed homotopy. Idea of the construction of 푋+. The idea of the construction of 푋+ is to attach 2-cells to 푋 to kill 푃 from 휋1(푋), then attach 3-cells to make sure that we retain the correct homology. Proofs can be found in [6, Prop. 4.40; 9, Thé. 1.1.1]. From this explicit construction, it is possible to use obstruction theory to show that 푋+ has the desired universal property, which in turn characterizes 푋+ uniquely up to homotopy equivalence. □ The plus construction is functorial in the following sense. perf 1.15. Corollary. Let CW∗ denote the category with objects pairs (푋, 푃), where 푋 is a pointed connected CW-complex and 푃 ⊲ 휋1(푋) is a perfect normal subgroup, and a morphism 푓∶ (푋, 푃) → (푋′, 푃′) ′ ′ is a map 푓∶ 푋 → 푋 so that 푓⋆(푃) ⊂ 푃 . The plus construction defines a functor + perf (−) ∶ CW∗ → ℎCW∗ . 1.16. Notation. For a commutative ring 푅, let 퐵GL(푅)+ denote the plus construction of 퐵GL(푅) with respect to the perfect normal subgroup 퐸(푅) ⊲ GL(푅) ≅ 휋1(퐵GL(푅)). 1.17. Definition. Let 푅 be a commutative ring and 푛 a positive integer. The 푛th algebraic 퐾-group of 푅 is the homotopy group + 퐾푛(푅) ≔ 휋푛(퐵GL(푅) ) . 1.18. Example. By the definition of the plus construction, for any ring 푅 we have 퐾1(푅) ≅ GL(푅)/퐸(푅) , so this definition agrees with the preliminary definition of 퐾1. 1.19. Warning. Notice that we have expressly avoided mentioning anything about 퐾0! That is because 퐾0 needs to be defined differently — Jesse will give a more careful analysis of 퐾0, 퐾1, and 퐾2 in his talk on 3 October. 4 PETER J. HAINE 2. A Whirlwind Tour of 퐾-theory In this section we give one manifestation of the connections between algebraic 퐾-theory and stable homotopy theory, via the Barratt–Priddy–Quillen theorem, as well as indicate how difficult 퐾-theory computations are by stating the the Kummer–Vandiver conjecture from number theory is equivalent to a 퐾-theory computation of the integers. + 2.1. Theorem (Barratt–Priddy–Quillen [3]). Let 퐵훴∞ denote the plus construction of 퐵훴∞ with respect to 퐴∞. There is an equivalence + ∞ ∞ 0 푛 푛 0 푛 푛 퐵훴∞ ≃ 훺 훴 푆 = colim푛 훺 훴 푆 = colim푛 훺 푆 . + 푆 In particular, 퐵훴∞ ≅ 휋푛. *2.1. Philosophical Remark. Now, 훴∞ is not the infinite general linear group of any ring, but if there were a field 퐅1 with one element, heuristically GL푛(퐅1) should be 훴푛, so GL(퐅1) 푆 should be 훴∞. Then under this heuristic, 퐾푛(퐅1) ≅ 휋푛. This is only philosophy, but later we will see that this can be made precise with the algebraic 퐾-theory of various types of 푆 categories. In this setting 퐾푛(Fin∗) ≅ 휋푛. 2.2. Observation (Relating algebraic 퐾-theory to stable homotopy theory). For each posi- tive integer 푛, there is an inclusion 훴푛 ↪ GL푛(퐙) given by the permutation representation, hence this induces an inclusion 훴∞ ↪ GL(퐙). The elementary integer matrices generate SL푛(퐙), and the matrix of an even permutation is 1, so we have compatible inclusions 퐴∞ SL(퐙) 훴∞ GL(퐙) , + + so by the functoriality of the plus construction we get an induced map 퐵훴∞ → 퐵GL(퐙) . Taking homotopy groups and using the Barratt–Priddy–Quillen theorem gives a map 푆 휋푛 → 퐾푛(퐙) . 2.3. Observation. Since 퐙 is the initial ring, every ring receives a unique map 휙푅 ∶ 퐙 → 푅.
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